Looking at the tournament as a whole (including the 5 nations), England have won the tournament most times, won most grand slams, and most triple crowns.

If you just look since 2000, they've also won the tournament most times.

Having said that, Wales have won the grand slam more times since it became the 6 nations. I think this is probably a greater achievement, especially when viewed against the relative success of England during this period.

Source: https://en.wikipedia.org/wiki/Six_Nations_Championship